Acceleration

When the speed changes

A bus pulls away from a stop: its speed grows. A cyclist brakes: her speed falls. Acceleration tells us how quickly the velocity changes:

Here $u$ is the starting velocity, $v$ the final velocity and $t$ the time taken. The unit is metres per second per second, $\text{m/s}^2$.

An acceleration of $2\ \text{m/s}^2$ means the velocity grows by $2\ \text{m/s}$ every second.

Speeding up and slowing down

If the velocity grows, $v > u$ and $a$ is positive. If it falls, $v < u$ and $a$ is negative. A negative acceleration is sometimes called a retardation or deceleration.

You should find that the acceleration turns negative as soon as the final velocity is below the start, and that the same change spread over a longer time gives a smaller acceleration.

Watch out. Zero velocity does not mean zero acceleration. A ball thrown up stops for a moment at the top, but gravity is still changing its velocity.

A worked example

A scooter speeds up from $4\ \text{m/s}$ to $16\ \text{m/s}$ in $6\ \text{s}$. A car brakes from $20\ \text{m/s}$ to rest in $5\ \text{s}$. Find each acceleration.

1. Scooter. $a = \dfrac{16\ \text{m/s} - 4\ \text{m/s}}{6\ \text{s}} = 2\ \text{m/s}^2$.
2. Car. $a = \dfrac{0\ \text{m/s} - 20\ \text{m/s}}{5\ \text{s}} = -4\ \text{m/s}^2$.
3. So the car's velocity falls by $4\ \text{m/s}$ each second.

Practise with new numbers

Try it in Python

The program works out both examples.

Remember these

• $a = \dfrac{v - u}{t}$, in $\text{m/s}^2$.
• Speeding up: $a$ positive. Slowing down: $a$ negative.
• Steady (uniform) acceleration: the velocity changes by the same amount every second.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A cyclist goes from rest to $6\ \text{m/s}$ in $3\ \text{s}$. What is her acceleration?

2. A train slows from $25\ \text{m/s}$ to $15\ \text{m/s}$ in $20\ \text{s}$. What is its acceleration?

3. A car moves at a steady $60\ \text{km/h}$ on a straight road. What is its acceleration?

4. What does an acceleration of $3\ \text{m/s}^2$ mean in words?